Mech. Sci., 4, 391–396, 2013
www.mech-sci.net/4/391/2013/
doi:10.5194/ms-4-391-2013
© Author(s) 2013. CC Attribution 3.0 License.
Mechanical
Sciences
Open Access
Vector model of the timing diagram of automatic machine
A. Jomartov
Institute Mechanics and Mechanical Engineering, Almaty, Kazakhstan
Correspondence to: A. Jomartov (legsert@mail.ru)
Received: 25 December 2012 – Revised: 11 October 2013 – Accepted: 28 November 2013 – Published: 11 December 2013
Abstract. In this paper a vector model of timing diagram of automatic machine is developed, which allows
us to solve a variety dynamic tasks by changing the parameters of timing diagram of its mechanisms. The
connection between the parameters of the timing diagram of automatic machine and equations of motion
mechanisms through functions of position and transfer functions of mechanisms is established. The vector
model of timing diagram can be used to optimize the timing diagrams of looms and polygraphic machines.
1
Introduction
Modeling of the timing diagram is one of the main parts
for design of automatic machines. A detailed analysis of the
works on the theory of the timing diagram performed prior
to 1965 is given in Petrokas (1970). Timing diagram is a
sequence of machine operations performed by mechanisms
depending on the angular displacement of the main shaft
(Browne, 1965; Youssef and El-Hofy, 2008; Homer, 1998;
Sandler, 1999; Natale C, 2003; Singh and Bhattacharya,
2006; Levner and Kats, 2007; Niir Board, 2009; Norton,
2009; Topalbekiroglu and Celik, 2009). Timing diagram allows determining of the position of each of the executive
body at any position of the main shaft (see Fig. 1).
Timing diagram automatic machine is modeled by a directed graph (see Fig. 2) (Novgorodtsev, 1982). The disadvantages of this model are the lack of consideration for connections executive bodies displacements mechanisms and accounting precision of manufacturing.
Analysis of the methods of synthesis and analysis of timing diagram automatic machine showed the need for further
development of optimization methods of timing diagram,
taking into account the accuracy of manufacture and the dynamics of automatic machine.
2
Vector model of the timing diagram of automatic
machine
The timing diagram of automatic machines can be represented as the vector polygons (Jomartov, 2010, 2011) (see
Published by Copernicus Publications.
Fig. 3). Let us replace the segments linear timing diagram by
the vectors `i j . The vectors `i j is directed sequentially from
one position to another position of mechanism, where i is
the number of mechanisms, j is the number of position of imechanism, m j is the number of positions of i-mechanism, n
is the number of mechanisms.
The projection of vectors `i j on the x axis is αi j – phase
angles of actuation of mechanisms. The projection `i j on the
y axis is the displacement δi j of j-position of i-mechanism.
δi j =
S ij
, S max = max S i j , i = 1, . . . , n; j = 1, . . . , mi ,
S max
where S i j is the displacement of j-position of i-mechanism.
To explain the parameters αi j and S i j in Fig. 4 shows a diagram of the displacement of mechanism, in the figure denote:
αo – phase angles of actuation of mechanism in the position
of open, αd is the phase angles of actuation of mechanism in
the position of dwell, αc is the phase angles of actuation of
mechanism in the position of close, S o is the displacement of
mechanism in the position of open, S c is the displacement of
mechanism in the position of close.
Let us introduce the vector P connecting the point of beginning and end of the cycle. The projection of the vector P
on the x axis is 2 π on the y axis is zero. The interaction of
mechanisms with each other will reflect in the form of the
vectors of connection cik , where k = 1, . . . , ri , ri is the number of vectors of connection of i-mechanism. The projection
of the vectors of connection to the x axis is the delay of actuation mechanism, and the projection on the y axis is the
difference between the displacements of mechanisms.
392
A. Jomartov: Vector model of the timing diagram of automatic machine
1
Figure
2. Presentation of timing diagram automatic machine as a
2
directed
graph.
3
Figure 2. Presentation of timing diagram automatic machine as a directed graph
1
2
Figure 1. Linear timing diagram of working and auxiliary cams of
a four-spindle bar automatic.
3
Figure 1. Linear timing diagram of working and auxiliary cams of a four-spindle bar
4
automatic.
5
Let us impose the timing diagram of mechanisms at each
other using zero vectors (see Fig. 3) connecting the boundary
points of timing diagram mechanisms in the y axis.
Let us compose the system of vector equations describing
the works of mechanisms automatic machine (see Fig. 3).
mi
P


`i j = P, i = 1, . . . , n, 




j=1
,

m
n
i

P P



bi j · `i j
cik =

(1)
i=1 j=1
where bi j ∈ {0, ± 1}.
Let us projected the vector Eq. (1) on the axis x and y.
mi
P
j=1
cikx
αi j = 2 π,
=
mi
n P
P
i=1 j=1
mi
P
j=1
δi j = 0,
bi j αi j , cyik =
mi
n P
P
i=1 j=1
bi j δi j













(2)
(3)
where αm
i j is the minimum allowable phase angles of actuation of mechanisms, δ`i j , δH
i j is the upper and lower limits
assigned by the designer.
On the projection vectors of connection impose constraints
y
yH
cikx` ≥ cikx ≥ cikx` , cy`
ik ≥ cik ≥ cik
(4)
y
y
y
x
where cikxH = eikx + ∆cikx , cyH
ik = eik + ∆cik where eik , eik are the
minimum permissible projection vectors of connection, ∆cikx ,
∆cyik are the errors of the projections of vectors of connection,
cikx` , cy`
ik are the upper limits imposed by the designer.
Equation (2) and constraints (Eqs. 3 and 4) describe the
collaboration works of mechanisms (timing diagram) of automatic machine.
Mech. Sci., 4, 391–396, 2013
A mathematical model of automatic machine
based on the timing diagram of mechanisms
Lets define the connection between the differential equations
of motion the automatic machine and the equations describing its timing diagram. In Fig. 5 shows the dynamic model
of the machine, where ci is the elasticity coefficients, βi is
the coefficients of resistance, Ji , Ii are the moments of inertia, M∂B is the motor torque, Mi is moment of resistance, Πi ,
i = 1, . . . , n is the function of position of mechanisms.
To compile the equations of motion mechanisms automatic machine (see Fig. 5), let us use Lagrange equations II
(Wolfson, 1976):

m
P
∂T
∂V
d ∂T


λi hi j 

dt ∂ϕ̇ j − ∂ϕ j + ∂ϕ j = Q j +


i=1
(5)
m+n

P


2

hi j ϕ̇ j + hi = 0,


j=1
On the phase angles of actuation of mechanisms αi j , and displacements of mechanisms δi j impose constraints
H
`
αi j ≥ αm
i j , δi j ≥ δi j ≥ δi j ,
3
where ϕ1 , ϕ2 , . . . , ϕn are the generalized coordinates, λi is
Lagrange multipliers, hi j , hi are some functions, T is the kinetic energy of a holonomic system, V is the potential energy
of the system, Q j is the generalized force.
To establish the connection between the equations of timing diagram (2–4) and the dynamic Eq. (5), let us write the
1
functions
of position, the transfer functions of mechanisms
of automatic machine in the following form:
"
!#
!
j−1
m
Pj
P
P
Πi = Πi1 · 1 − L (φi − αi1 ) +
Πi j 1 − L φi −
αir · L φi −
αir
j=2
r=1
r=1
"
!#
!
j−1
m
Pj
P
P
Π0i = Π0i1 1 − L (φi − αi1 ) +
Π0i j 1 − L φi −
αir · L φi −
αir
j=2
r=1
r=1
"
!#
!
j−1
m
Pj
P
P
Π00i = Π00i1 1 − L (φi − αi1 ) +
Π00i j 1 − L φi −
αir · L φi −
αir
j=2
r=1
r=1














(6)













where i = 1,. . . . , n, L(x) is a step function of the form
(
0, x < 0,
L(x) =
.
1, x ≥ 0
Πi j , Π0i j , Π00i j are the functions of position, the first transfer
function, the second transfer function on parts of phase angles of actuation αi j of mechanisms.
Equation (6) establish a connection between the Eq. (5)
describe the dynamics of the automatic machine and the
Eqs. (2)–(4) timing diagram of the machine-automaton. This
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A. Jomartov: Vector model of the timing diagram of automatic machine
393
Figure 3. Vector
1 model of the timing diagram of automatic machine.
2
3
d Πi (φi )
d2 Πi (φi )
Figure 3. Vector model of the timing diagram of automatic
Π0i (φi ) =machine. ; Π00i (φi ) =
; i = 1, 2.
2
d φi
Figure 7 shows vector timing diagram of automatic machine,
which is described by the following equations:

l11 + l12 = P 



l21 + l22 = P 
.
(8)


c =l −l 
1
2
Figure 4. Diagram of the displacement of mechanism.
3
Figure 4. Diagram of the displacement of mechanism.
21
method allows to solve various optimization tasks, where the
variable parameters are the phase angles αi j and of the displacements δi j of timing diagram automatic machine.
4
An example
Let us show in more detail the connection between timing
diagram of automatic machine and dynamics of mechanisms
on the example of automatic machine with two cam mechanisms. The dynamic model is shown in Fig. 6, where ϕ0 , ϕ1 ,
ϕ2 are generalized coordinates, MD is the motor torque, M1 ,
M2 are the moments of resistance, I1 , J0 , J1 , J2 are the moments of inertia of mechanisms, c0 , c1 are the coefficients of
elasticity of shafts, Πi (φi ) is the function of position of cam
mechanisms, Π0i (φi ) is the first transfer functions of the cams,
Π00i (φi ) is the second transfer functions of the cams.
This dynamic model is described by the following
equations
J0 φ̈0 + c1 (φ0 0 − φ 1 ) = MD ,
J + I Π2 (φ ) φ̈ + I Π0 (φ ) Π00 (φ ) φ̇2 + c (φ − φ0 ) + c1 (φ1 − phi2 ) = −M1 Π01 (φ1 ) ,
1 1 10 1 1 1 1 1 1 1 1 0 1
J2 + I2 Π22 (φ2 ) φ̈2 + I2 Π02 (φ2 ) Π002 (φ2 ) φ̇22 + c2 (φ2 − φ2 ) = −M2 Π20 (φ2 )
where
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d φi






(7)





21
11
Let us projected Eq. (8) on the x, y respectively

α11 + α12 = 2 π 



α21 + α22 = 2 π 


x
c11 = α21 − α11 

δ11 − δ12 = 0 



δ21 − δ22 = 0 
.


y
c =δ −δ 
21
11
(9)
(10)
11
Impose the constraints on the phase angles, displacements of
mechanisms, and projections of vectors of connection
αi j ≥ αmin
ij
δmax
≥ δi j ≥ δmin
ij
ij
xmax
x
xmin
c11
≥ c11
≥ c11
ymax
y
ymin
c11 ≥ c11 ≥ c11








.







(11)
3
The expressions (Eqs. 9–11) allow you to vary the phase
angles and displacements of mechanisms of automatic machine, without disrupting their normal work.
To establish the connection between the equations describing of joint work of the mechanisms of automatic machine
(Eqs. 9–11) and the dynamic Eq. (7), let us write the functions of position and the transfer functions of mechanisms of
automatic machine (see Fig. 8) as follows:
4
Mech. Sci., 4, 391–396, 2013
394
A. Jomartov: Vector model of the timing diagram of automatic machine
Figure 5. A1dynamic model of automatic machine.
2
3
Figure 5. A dynamic model of automatic machine.
1
Figure 7. Vector timing diagram of automatic machine with two
2
cams7.mechanisms.
Figure
Vector timing diagram of automatic machine with two cams mechanisms.
3
1
Figure 6. Dynamic model of automatic machine with two cams
2
mechanisms.
3
Figure 6. Dynamic model of automatic machine with two cams mechanisms.
Πi = Πi1 · 1 − L (φi − αi1 ) + Πi2 1 − L (φi − (αi1 + αi2 )) · L (φi − αi1 )
Π0i = Π0i1 · 1 − L (φi − αi1 ) + Π0i2 1 − L (φi − (αi1 + αi2 )) · L (φi − αi1 )
Π00i = Π00i1 · 1 − L (φi − αi1 ) + Π00i2 1 − L (φi − (αi1 + αi2 )) · L (φi − αi1 )
i = 1, 2
Πi j = ai j (ki ) δi j
δ
Π0i j = bi j (ki ) αii jj









Π00i j = di j (ki )
(12)
L(x) =
0, x < 0,
.
1, x ≥ 0.
Represent the generalized coordinates ϕ1 , ϕ2 through the dimensionless coefficients k1 , k2 where ϕ1 = 2 π k1 ; ϕ2 = 2 π k2 ;
k1 , k2 ∈ [0, 1]
Mech. Sci., 4, 391–396, 2013
(13)
i = 1, 2; j = 1, 2
where
(
δi j
α2i j










.



6
ai j (ki ), bi j (ki ), di j (ki ); i = 1, 2; j = 1, 2; are coefficients of the
displacement, the velocity, the acceleration
5 of mechanism in
j-position.
Equations (12) and (13) establish a connection between the
phase angles of actuation of mechanisms αi j and magnitude
of displacement of mechanisms δi j and of their functions of
position and transfer functions that are explicitly included in
7
the equations of motion of the automatic machine (Eq. 7).
Now, depending on the chosen optimization criterion, by
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A. Jomartov: Vector model of the timing diagram of automatic machine
395
1
Figure 8. The functions
of position and the transfer functions of mechanisms of automatic machine.
2
3
Figure 8. The functions of position and the transfer functions of mechanisms of
Edited by: A. Barari
Reviewed by: two anonymous referees
varying the parameters of timing diagram αi j and δi j , can
improve the dynamics
of themachine.
automatic machine. As an opti4
automatic
mization criterion can use the following expression:
max Π0i (ϕi ) Π00i (ϕi ) .
ϕi
5
Conclusions
The vector model of timing diagram is developed on the basis
of representation timing diagram of the automatic machine as
vector polygons, which allows solving various dynamic tasks
at the expense of change of parameters of timing diagram.
The mathematical model of the automatic machine with
elastic links on the basis of the timing diagram of its mechanisms is received.
The equations of connection between parameters timing
diagram of the automatic machine and equations of dynamics through functions of position and transfer functions of
mechanisms were received.
The model of timing diagram of the automatic machine is
the only method which allows to solve a variety of dynamic
tasks, by optimizing its timing diagrams, at this time.
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